A Hybrid Method to Improve Forecasting Accuracy 2_3_5.pdf · 94 A Hybrid Method to Improve Forecasting Accuracy For these equations, recursive algorithm has been developed. In this

Journal of Computations & Modelling, vol.2, no.3, 2012, 89-108

ISSN: 1792-7625 (print), 1792-8850 (online)

Scienpress Ltd, 2012

A Hybrid Method to Improve Forecasting

Accuracy

- An Introduction of a Day of the Week Index for Air

Cargo Weight Data -

Tatsuhiro Kuroda1, Keiko Nagata

2 and Kazuhiro Takeyasu

3

Abstract

Air cargo loading weight forecasting is an important factor for managers in the

aviation industry because revenue is dependent on the amount of weight loaded. In

this paper, we propose a new method to improve forecasting accuracy and confirm

them by the numerical example. Focusing that the equation of exponential

smoothing method(ESM) is equivalent to (1,1) order ARMA model equation, a

new method of estimation of smoothing constant in exponential smoothing

method is proposed before by us which satisfies minimum variance of forecasting

error. Generally, smoothing constant is selected arbitrarily. But in this paper, we

utilize above stated theoretical solution. Firstly, we make estimation of ARMA

1 Student of Graduate School of Economics of Osaka Prefecture University in Japan,

e-mail: [email protected] 2 Student of Graduate School of Economics of Osaka Prefecture University in Japan,

e-mail: [email protected] 3 Fuji-Tokoha University, e-mail: [email protected]

Article Info: Received : September 12, 2012. Revised : October 8, 2012

Published online : November 20, 2012

mailto:[email protected]

90 A Hybrid Method to Improve Forecasting Accuracy

model parameter and then estimate smoothing constants. Thus theoretical solution

is derived in a simple way and it may be utilized in various fields. Combining the

trend removing method with this method, we aim to improve forecasting accuracy.

Furthermore, “a day of the week index” is newly introduced for the daily air cargo

weight data and we have obtained good result. The effectiveness of this method

should be examined in various cases.

Keywords: Minimum Variance, Exponential Smoothing Method, Forecasting,

Trend

1 Introduction

Many methods for time series analysis have been presented such as

Autoregressive model (AR Model), Autoregressive Moving Average Model

(ARMA Model) and Exponential Smoothing Method (ESM) (Box Jenkins [1]),

(R.G.Brown [11]), (Tokumaru et al. [3]), (Kobayashi [7]). Among these, ESM is

said to be a practical simple method.

For this method, various improving method such as adding compensating

item for time lag, coping with the time series with trend (Peter [10]), utilizing

Kalman Filter (Maeda [4]), Bayes Forecasting (M.West et al. [8]), adaptive ESM

(Steinar [13]), exponentially weighted Moving Averages with irregular updating

periods (F.R.Johnston [2]), making averages of forecasts using plural method

(Spyros [12]) are presented. For example, Maeda [4] calculated smoothing

constant in relationship with S/N ratio under the assumption that the observation

noise was added to the system. But he had to calculate under supposed noise

because he couldn’t grasp observation noise. It can be said that it doesn’t pursue

optimum solution from the very data themselves which should be derived by those

estimation. Ishii [9] pointed out that the optimal smoothing constant was the

Tatsuhiro Kuroda, Nagata Keiko and Kazuhiro Takeyasu 91

solution of infinite order equation, but he didn’t show analytical solution. Based

on these facts, we proposed a new method of estimation of smoothing constant in

ESM before (Takeyasu et al. [6]). Focusing that the equation of ESM is equivalent

to (1,1) order ARMA model equation, a new method of estimation of smoothing

constant in ESM was derived.

In this paper, utilizing above stated method, revised forecasting method is

proposed. In making forecast such as air cargo weight data, trend removing

method is devised. Trend removing by the combination of linear and 2nd order

non-linear function and 3rd order non-linear function is executed to the original air

cargo weight data. The weights for these functions are varied by 0.01 increment

and optimal weights are searched.

“a day of the week index (DWI)” is newly introduced for the daily data and a

day of the week trend is removed. Theoretical solution of smoothing constant of

ESM is calculated for both of the DWI trend removing data and the non DWI

trend removing data. Then forecasting is executed on these data. This is a revised

forecasting method. Variance of forecasting error of this newly proposed method

is assumed to be less than those of previously proposed method. The rest of the

paper is organized as follows. In Section 2, ESM is stated by ARMA model and

estimation method of smoothing constant is derived using ARMA model

identification. The combination of linear and non-linear function is introduced for

trend removing in Section 3. “a day of the week index (DWI)” is newly

introduced in Section 4. Forecasting is executed in Section 5, and estimation

accuracy is examined.

2 Description of ESM using ARMA model

In ESM, forecasting at time 1t is stated in the following equation.


Here,

:ˆ1tx

forecasting at 1t

:tx realized value at t

: smoothing constant 0 1

(1) is re-stated as:

By the way, we consider the following (1,1) order ARMA model.

Generally, qp, order ARMA model is stated as:

Here,

:tx Sample process of Stationary Ergodic Gaussian Process tx ,,,2,1 Nt

te :Gaussian White Noise with 0 mean 2

e variance

MA process in (4) is supposed to satisfy convertibility condition.

Utilizing the relation that:

we get the following equation from (3).

Operating this scheme on t +1, we finally get:

tt

tttt

xx

xxxx

ˆ1

ˆˆˆ1

(1)

lt

l

l

t xx

0

1 1ˆ (2)

11 tttt eexx (3)

jt

q

j

jtit

p

i

it ebexax

11

(4)

0,, 21 ttt eeeE

11ˆ

ttt exx (5)


If we set 1 , the above equation is the same with (1), i.e., equation of ESM

is equivalent to (1,1) order ARMA model, or is said to be (0,1,1) order ARIMA

model because 1st order AR parameter is 1 (Box Jenkins [1]), (Tokumaru et

al.[3]).

Comparing with (3) and (4), we obtain:

From (1), (6),

Therefore, we get:

From above, we can get estimation of smoothing constant after we identify the

parameter of MA part of ARMA model. But, generally MA part of ARMA model

become non-linear equations which are described below.

Let (4) be:

We express the autocorrelation function of tx as kr and from (8), (9), we get the

following non-linear equations which are well known [3].

ttt

ttt

xxx

exx

ˆ1ˆ

1ˆˆ1

(6)

1

1

1a

b

1

1

1

1

1

a

b

(7)

it

p

i

itt xaxx

1

~ (8)

jt

q

j

jtt ebex

1

~ (9)


For these equations, recursive algorithm has been developed. In this paper,

parameter to be estimated is only 1b , so it can be solved in the following way.

From (3) (4) (7) (10), we get:

If we set:

the following equation is derived.

We can get 1b as follows.

In order to have real roots, 1 must satisfy:

From invertibility condition, 1b must satisfy:

q

j

je

jk

kq

j

jek

br

qk

qkbbr

0

22

0

0

2

~

)1(0

)(~

(10)

2

11

22

10

1

1

~

1~

1

1

1

e

e

br

br

b

a

q

(11)

0

kk

r

r

(12)

11 2

11

b

b

(13)

2

1

1

1

1 1 4

2b

(14)

1

1

2 (15)


From (13), using the next relation,

(15) always holds.

As

1b is within the range of:

Finally we get:

which satisfy above condition. Thus we can obtain a theoretical solution by a

simple way.

Here 1 must satisfy:

in order to satisfy 0 1 .

Focusing on the idea that the equation of ESM is equivalent to (1,1) order ARMA

model equation, we can estimate smoothing constant after estimating ARMA

model parameter.

It can be estimated only by calculating 0th and 1st order autocorrelation function.

1 1b

01

01

2

1

2

1

b

b

1 1b

11 0b

2

1

1

1

2

1 1

1

1 1 4

2

1 2 1 4

2

b

(16)

1

10

2 (17)


3 Trend Removal method

As trend removal method, we describe the combination of linear and non-linear

function.

[1] Linear function

We set:

as a linear function.

[2] Non-linear function

We set:

as a 2nd

and a 3rd

order non-linear function.

[3] The combination of linear and non-linear function

We set:

as the combination of linear and 2nd

order non-linear and 3rd

order non-linear

function.

Trend is removed by dividing the data by (21).

4 A Day of the Week Index

“a day of the week index (DWI)” is newly introduced for the daily data. It is

often seen that cargo steadily increases from Monday to Saturday because most

1 1y x b (18)

2

2 2 2y x b x c (19)

3 2

3 3 3 3y x b x c x d (20)

　

　

33

2

3

3

33

22

2

22111

dxcxbxa

cxbxabxay

(21)

1 2 3

1 2 3

0 1, 0 1, 0 1

1

(22)


factories start manufacturing products from Monday. Then they are delivered to

airlines as cargo. Therefore a day of the week index may be considered and the

forecasting accuracy would improve after we identify the “a day of the week

index” and utilize them. This time in this paper, the data we handle consist by

Monday through Sunday, we calculate jDWI 7,,1 j for Monday through

Sunday.

For example, if there is the daily data of L weeks as stated bellow:

where ijx R in which L means the number of weeks (Here 10L ), i

means the order of weeks ( i -th week), j means the order in a week ( j -th order

in a week; for example 1j : Monday, 7j : Sunday) and ijx is air cargo

weight data. Then, jDWI is calculated as follows.

DWI trend removal is executed by dividing the data by (23). Numerical

examples both of DWI removal case and non-removal case are discussed in

Section 5.

5 Forecasting the air cargo weight data

5.1 Analysis Procedure

The air cargo weight data of 4 cases from April 1, 2012 (Sun.) to June 23,

2012 (Sat.) are analyzed. First of all, graphical charts of these time series data are

exhibited in Figure 1 to 4.

7,,1,,1 jLixij

L

i j

ij

L

i

ij

j

xL

xL

DWI

1

7

1

1

7

11

1

(23)


Figure 1: Air cargo weight data of flight rout A

Figure 2: Air cargo weight data of flight rout B

Figure 3: Air cargo weight data of flight rout C


Figure 4: Air cargo weight data of flight rout D

Analysis procedure is as follows. There are 84 daily data for each case. We

use 70 data(1 to 70) and remove trend by the method stated in Section 3. Then we

calculate a day of the week index (DWI) by the method stated in Section 4. After

removing DWI trend, the method stated in Section 2 is applied and Exponential

Smoothing Constant with minimum variance of forecasting error is estimated.

Then 1 step forecast is executed. Thus, data is shifted to 2nd to 71st and the

forecast for 72nd data is executed consecutively, which finally reaches forecast of

84th data. To examine the accuracy of forecasting, variance of forecasting error is

calculated for the data of 71st to 84th data. Final forecasting data is obtained by

multiplying DWI and trend.

Forecasting error is expressed as:

Variance of forecasting error is calculated by:

In this paper, we examine the two cases stated in Table 1.

iii xx ˆ (24)

N

i

iN

1

1 (25)

2

1

2

1

1

N

i

iN

(26)


Table 1: The combination of the case of trend removal and DWI trend removal

Case Trend DWI trend

Case1 Removal Removal

Case2 Removal Non removal

5.2 Trend removing

Trend is removed by dividing original data by (21). Here, the weight of 1

and 2 are shifted by 0.01 increment in (21) which satisfy the equation (22). The

best solution is selected which minimizes the variance of forecasting error.

Estimation results of coefficient of (18), (19) and (20) are exhibited in Table 2.

Data are fitted to (18), (19) and (20), and using the least square method,

parameters of (18), (19) and (20) are estimated. Estimation results of weights of

(21) are exhibited in Table 3. The weighting parameters are selected so as to

minimize the variance of forecasting error.

Table 2: Coefficient of (18),(19) and (20)

1st 2

nd 3

rd

Rout 1a 1b 2a 2b 2c 3a 3b 3c 3d

A 32.1 9,886.8 -2.1 183.6 7,868.8 0 -2.2 184.3 7,864.2

B 31.5 8,106.8 -1.0 102.8 7,251.3 -0.1 6.8 -119.2 8,610.8

C 71.0 5,862.9 0.1 66.8 5,913.9 0 2.5 -2.8 6,340.2

D -57.3 8,462.3 0 -53.8 8,421.0 0.1 -6.7 135.6 7,260.8


Table 3: Weights of (21)

Rout Case 1 2 3

A Case1 0.70 0.30 0.00

Case2 1.00 0.00 0.00

B Case1 0.00 0.00 1.00

Case2 0.00 0.00 1.00

C Case1 0.00 0.64 0.36

Case2 0.59 0.00 0.41

D Case1 0.71 0.06 0.23

Case2 0.93 0.01 0.06

As a result, we can observe the following six patterns.

① Selected liner model:

Rout A Case2

② Selected 3rd order model:

Rout B Case1/2

③ Selected 1st and 2nd model:

Rout A Case1

④ Selected 1st and 3rd model:

Rout C Case2

⑤ Selected 2nd and 3rd mode;

Rout C Case1

⑥ Selected 1st, 2nd and 3rd model:

Rout D Case1/2

Graphical charts of trend are exhibited in Figure 5 to 8.


Figure 5: Trend of flight rout A

Figure 6: Trend of flight rout B

Figure 7: Trend of flight rout C


Figure 8: Trend of flight rout D

5.3 Removing Trend by DWI

After removing trend, a day of the week index is calculated by the method

stated in 4. Calculation result for 1st to 70th data is exhibited in Table 4.

Table 4: a day of the week index

Rout Case a day of the week index

Sun. Mon. Tue. Wed. Thu. Fri. Sat.

A Case1 1.117 0.889 0.787 1.052 0.953 1.023 1.179

B Case1 0.967 1.135 1.070 0.879 0.820 1.137 0.993

C Case1 1.128 1.048 0.963 0.955 1.021 0.878 1.008

D Case1 1.066 1.110 0.930 0.881 0.889 1.167 0.957


5.4 Estimation of Smoothing Constant with Minimum Variance of

Forecasting Error

After removing DWI trend, Smoothing Constant with minimum variance of

forecasting error is estimated utilizing (16). There are cases that we cannot obtain

a theoretical solution because they do not satisfy the condition of (17). In those

cases, Smoothing Constant with minimum variance of forecasting error is derived

by shifting variable from 0.01 to 0.99 with 0.01 interval. Calculation result for 1st

to 70th data is exhibited in Table 5.

Table 5: Estimated Smoothing Constant with Minimum Variance

Rout Case 1

A Case1 -0.40097 0.49797

Case2 -0.36307 0.56970

B Case1 -0.45484 0.35727

Case2 -0.42518 0.44282

C Case1 -0.73855 0.16000

Case2 -0.47261 0.28740

D Case1 -0.13000 0.86773

Case2 -0.13643 0.86093

5.5 Forecasting and variance of forecasting error

Utilizing smoothing constant estimated in the previous section, forecasting is

executed for the data of 71st to 84th data. Final forecasting data is obtained by

multiplying DWI and trend. Variance of forecasting error is calculated by (26).

Forecasting results are exhibited in Figure 9 to 12.


Figure 9: Forecasting Results of flight rout A

Figure 10: Forecasting Results of flight rout B

Figure 11: Forecasting Results of flight rout C


Figure 12: Forecasting Results of flight rout D

Variance of forecasting error is exhibited in Table 6.

Table 6: Variance of Forecasting Error

Rout Case Variance of Forecasting Error

A Case1 1,589,326.7 *

Case2 3,031,109.2

B Case1 4,200,368.1 *

Case2 4,437,768.8

C Case1 3,061,739.7 *

Case2 4,502,923.4

D Case1 3,757,962.0 *

Case2 3,817,648.5

6 Conclusions

Focusing on the idea that the equation of exponential smoothing


method(ESM) was equivalent to (1,1) order ARMA model equation, a new

method of estimation of smoothing constant in exponential smoothing method was

proposed before by us which satisfied minimum variance of forecasting error.

Generally, smoothing constant was selected arbitrarily. But in this paper, we

utilized above stated theoretical solution. Firstly, we made estimation of ARMA

model parameter and then estimated smoothing constants. Thus theoretical

solution was derived in a simple way and it might be utilized in various fields.

Furthermore, combining the trend removal method with this method, we

aimed to increase forecasting accuracy. An approach to this method was executed

in the following method. Trend removal by a linear function was applied to the

loading weight data of air cargo. The combination of linear and non-linear

function was also introduced in trend removing. “a day of the week index (DWI)”

is newly introduced for the daily data and a day of the week trend is removed.

Theoretical solution of smoothing constant of ESM was calculated for both of the

DWI trend removing data and the non DWI trend removing data. Then forecasting

was executed on these data.

In all cases, Case1 (DWI is imbedded) is better than Case 2 (DWI is not

imbedded). In particular, Flight rout A obtained almost double twice improvement

in forecasting accuracy. Flight rout C had also the good result.

The effectiveness of this method should be examined in various cases.

References

[1] Box Jenkins, Time Series Analysis, Third Edition, Prentice Hall, 1994.

[2] F.R. Johnston, Exponentially Weighted Moving Average (EWMA) with

Irregular Updating Periods, Journal of the Operational Research Society,

44(7), (1993), 711-716.

[3] Hidekatsu Tokumaru et al., Analysis and Measurement–Theory and


Application of Random data Handling, Baifukan Publishing, 1982.

[4] Katsuro Maeda, Smoothing Constant of Exponential Smoothing Method,

Seikei University Report Faculty of Engineering, 38, (1984), 2477-2484.

[5] Kazuhiro Takeyasu, System of Production, Sales and Distribution

Chuokeizai-Sha Publishing, 1996.

[6] Kazuhiro Takeyasu and Kazuko Nagao, Estimation of Smoothing Constant

of Minimum Variance and its Application to Industrial Data, Industrial

Engineering and Management Systems, 7(1), (2008), 44-50.

[7] Kengo Kobayashi, Sales Forecasting for Budgeting, Chuokeizai-Sha

Publishing, 1992.

[8] M. West and P.J. Harrison, Baysian Forecasting and Dynamic Models,

Springer-Verlag, New York, 1989.

[9] Naohiro Ishii et al., Bilateral Exponential Smoothing of Time Series, Int. J.

System Sci., 12(8), (1991), 997-988.

[10] Peter R. Winters, Forecasting Sales by Exponentially Weighted Moving

Averages, Management Science, 6(3), (1984), 324-343.

[11] R.G. Brown, Smoothing, Forecasting and Prediction of Discrete–Time Series,

Prentice Hall, 1963.

[12] Spyros Makridakis and Robeat L. Winkler, Averages of Forecasts; Some

Empirical Results, Management Science, 29(9), (1983), 987-996.

[13] Steinar Ekern, Adaptive Exponential Smoothing Revisited, Journal of the

Operational Research Society, 32, (1982), 775-782.

Documents

A Hybrid Method to Improve Forecasting Accuracy 2_3_5.pdf · 94 A Hybrid Method to Improve Forecasting Accuracy For these equations, recursive algorithm has been developed. In this